We will discuss the question of defining a p-adic L-function and formulating a main conjecture for an Artin representation. The case where the Artin representation is totally even (or odd) is classical. The corresponding main conjecture has been proven by Wiles. This talk will discuss the special case where the representation is 2-dimensional, but not totally even or odd. As we will explain, under certain assumptions, there are two p-adic L-functions, two Selmer groups, and two main conjectures. This talk is about joint work with Nike Vatsal.